Wickets in 3-uniform hypergraphs

In this note, we consider a Tur & aacute;n-type problem in hypergraphs. What is the maximum number of edges if we forbid a subgraph? Let H ( 3 ) n be a 3-uniform linear hypergraph, i.e. any two edges have at most one vertex common. A special hypergraph, called wicket , is formed by three rows and two columns of a 3 x 3 point matrix. We describe two linear hypergraphs that if we forbid either of them in H n ( 3 ) , then the hypergraph is sparse, i.e. the number of its edges is o ( n 2 ) . Since both contain a wicket, it implies a conjecture of Gy & aacute;rf & aacute;s and S & aacute;rk & ouml;zy that wicket-free hypergraphs are sparse. (c) 2024 Elsevier B.V. All rights reserved.